Load libraries

library(tidyverse) # data manipulation
library(ggpubr) # producing data exploratory plots
library(modelsummary) # descriptive data 
library(glmmTMB) # running generalised mixed models 
library(DHARMa) # model diagnostics 
library(performance) # model diagnostics  
library(ggeffects) # partial effect plots 
library(car) # running Anova on model 
library(emmeans) # post-hoc analysis

Import data

m1 <- read_csv("import_data/1_month_size_data_2022_2023.csv") |> 
  mutate(across(1:15,factor)) |> 
  mutate(STANDARD_LENGTH =LENGTH, 
         .keep = "unused") |> 
  select(!(NOTES)) |> 
  select(1:15,"STANDARD_LENGTH","MASS")
           
m2 <- read_csv("import_data/2_month_size_data_2022_2023.csv") |> 
  mutate(across(1:15,factor)) |> 
  mutate(STANDARD_LENGTH=LENGTH, 
         .keep = "unused") |> 
  select(!(NOTES)) |> 
  select(1:15,"STANDARD_LENGTH","MASS")
m2.5 <- read_csv("import_data/2-5_month_size_data_2022_2023.csv") |> 
  mutate(across(1:15,factor)) |> 
  mutate(STANDARD_LENGTH =LENGTH, 
         .keep = "unused") |> 
  select(!(NOTES)) |> 
  select(1:15,"STANDARD_LENGTH","MASS")
adult <- read_csv("import_data/adult_size_2022_2023.csv") |> 
  mutate(across(1:3,factor), 
         MALE = FISH_ID, 
         FEMALE = FISH_ID, 
         POPULATION = str_sub(FISH_ID, 2,4), 
         POPULATION = case_when(POPULATION == "ARL" ~ "Arlington Reef", 
                                POPULATION == "SUD" ~ "Sudbury Reef",
                                POPULATION == "VLA" ~ "Vlassof cay",
                                POPULATION == "PRE" ~ "Pretty patches", 
                                TRUE ~ POPULATION)) |> 
  left_join(select(m1, c("MALE","TEMPERATURE")), 
             by="MALE") |> 
  left_join(select(m1, c("FEMALE","TEMPERATURE")), 
             by="FEMALE") |>
  distinct() |> 
  mutate(TEMPERATURE = coalesce(TEMPERATURE.x, TEMPERATURE.y)) |> 
  drop_na(TEMPERATURE) |> 
  select(-c("TEMPERATURE.x","TEMPERATURE.y"))

Data manipulation

m1_df <- m1 |> 
  left_join(select(adult, c("MALE", "SL", "MASS")), 
            by ="MALE") |> 
  mutate(SL_MALE =SL, 
         MASS_MALE =MASS.y, 
         .keep = "unused") |>
  left_join(select(adult, c("FEMALE", "SL", "MASS")), 
            by ="FEMALE") |> 
  mutate(SL_FEMALE =SL, 
         MASS_FEMALE =MASS, 
         .keep ="unused") |> 
  mutate(SL_MIDPOINT = (SL_MALE+SL_FEMALE)/2, 
         MASS_MIDPOINT = (MASS_MALE+MASS_FEMALE)/2)

m2_df <- m2 |> 
  left_join(select(adult, c("MALE", "SL", "MASS")), 
            by ="MALE") |> 
  mutate(SL_MALE =SL, 
         MASS_MALE =MASS.y, 
         .keep = "unused") |>
  left_join(select(adult, c("FEMALE", "SL", "MASS")), 
            by ="FEMALE") |> 
  mutate(SL_FEMALE =SL, 
         MASS_FEMALE =MASS, 
         .keep ="unused") |> 
  mutate(SL_MIDPOINT = (SL_MALE+SL_FEMALE)/2, 
         MASS_MIDPOINT = (MASS_MALE+MASS_FEMALE)/2)

m2.5_df <- m2.5 |> 
  left_join(select(adult, c("MALE", "SL", "MASS")), 
            by ="MALE") |> 
  mutate(SL_MALE =SL, 
         MASS_MALE =MASS.y, 
         .keep = "unused") |>
  left_join(select(adult, c("FEMALE", "SL", "MASS")), 
            by ="FEMALE") |> 
  mutate(SL_FEMALE =SL, 
         MASS_FEMALE =MASS, 
         .keep ="unused") |> 
  mutate(SL_MIDPOINT = (SL_MALE+SL_FEMALE)/2, 
         MASS_MIDPOINT = (MASS_MALE+MASS_FEMALE)/2) |> 
  drop_na(SL_MALE)

Exploratory data analysis

MASS

plot1 <- ggplot(m1_df, aes(x=MASS_MALE, y=MASS.x, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") +
  ylim(0,0.15) +
  theme_classic()

plot2 <- ggplot(m1_df, aes(x=MASS_FEMALE, y=MASS.x, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") + 
  ylim(0,0.15) +
  theme_classic()

plot3 <- ggplot(m1_df, aes(x=MASS_MIDPOINT, y=MASS.x, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") + 
  ylim(0,0.15) +
  theme_classic()

ggarrange(plot1, plot2, plot3, 
          nrow =1, 
          ncol =3, 
          common.legend = TRUE)

STANDARD_LENGTH

plot1 <- ggplot(m1_df, aes(x=SL_MALE, y=STANDARD_LENGTH, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") + 
  theme_classic()

plot2 <- ggplot(m1_df, aes(x=SL_FEMALE, y=STANDARD_LENGTH, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") + 
  theme_classic()

plot3 <- ggplot(m1_df, aes(x=SL_MIDPOINT, y=STANDARD_LENGTH, color=TEMPERATURE)) +
  geom_point(alpha=0.05) + 
  stat_smooth(method = "lm") + 
  theme_classic()

ggarrange(plot1, plot2, plot3, 
          nrow =1, 
          ncol =3, 
          common.legend = TRUE)

Descriptive statistics

counts

Adults

datasummary(Factor(POPULATION) ~ Factor(TEMPERATURE), 
            data=adult, 
            fmt = "%.0f")
tinytable_y79w3dvh4h9lgoxrg5np
POPULATION 27 28.5 30
Arlington Reef 8 8 8
Pretty patches 4 6 6
Sudbury Reef 4 4 2
Vlassof cay 6 2 6

1-months

datasummary(Factor(POPULATION) ~ Factor(TEMPERATURE), 
            data=m1_df, 
            fmt = "%.0f")
tinytable_kzl9ejyu3sutdi4easud
POPULATION 27 28.5 30
Arlington Reef 231 105 202
Pretty Patches 116 82 142
Sudbury Reef 117 55 47
Vlassof Cay 114 60 120

2-months

datasummary(Factor(POPULATION) ~ Factor(TEMPERATURE), 
            data=m2_df, 
            fmt = "%.0f")
tinytable_n43cuyql93yglx2idq49
POPULATION 27 28.5 30
Arlington Reef 198 108 152
Pretty Patches 113 83 134
Sudbury Reef 113 60 56
Vlassof Cay 77 58 113

2.5-months

datasummary(Factor(POPULATION) ~ Factor(TEMPERATURE), 
            data=m2.5_df, 
            fmt = "%.0f")
tinytable_gvs61g8hf8ivrl7kgcqh
POPULATION 27 28.5 30
Arlington Reef 239 103 211
Pretty Patches 100 83 133
Sudbury Reef 111 53 50
Vlassof Cay 102 60 106

standard length

Adults

datasummary(Factor(TEMPERATURE) ~ MASS * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(adult, MASS),  
            fmt = "%.2f")
tinytable_eulkhl22rjhdx0jk7v2m
TEMPERATURE NUnique mean median min max sd Histogram
27 21 46.93 50.63 29.85 59.93 10.30 ▇▁▃▄▄▆▄
28.5 20 38.81 42.36 16.28 53.26 10.26 ▁▁▄▃▄▇▆▁
30 22 39.94 39.62 23.91 57.31 9.43 ▅▇▂▇▇▇▂▇▅▂

1-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m1_df, MASS.x),  
            fmt = "%.2f")
tinytable_paqv1109n6yanf6yrqsf
TEMPERATURE NUnique mean median min max sd Histogram
27 437 0.06 0.06 0.01 0.22 0.03 ▃▇▇▄▂▁
28.5 248 0.07 0.06 0.02 0.19 0.02 ▁▅▇▄▃
30 389 0.07 0.07 0.01 0.15 0.02 ▄▅▇▆▅▃▂▁

2-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m2_df, MASS.x),  
            fmt = "%.2f")
tinytable_hyamk4t01lfqfd4ehfv1
TEMPERATURE NUnique mean median min max sd Histogram
27 460 0.27 0.27 0.03 0.68 0.10 ▁▂▄▇▆▃▁
28.5 297 0.28 0.27 0.06 0.64 0.10 ▁▃▇▇▇▄▁▁▁
30 425 0.28 0.27 0.04 0.62 0.11 ▂▂▅▇▇▄▂▁▁

2.5-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m2.5_df, MASS.x),  
            fmt = "%.2f")
tinytable_fc2lhueoxgm2og9pbghc
TEMPERATURE NUnique mean median min max sd Histogram
27 529 0.46 0.46 0.04 1.00 0.17 ▁▂▄▇▇▆▄▁▁
28.5 293 0.45 0.44 0.08 1.00 0.18 ▂▄▆▇▆▅▃▂▁
30 477 0.45 0.45 0.08 0.96 0.15 ▁▂▄▆▇▄▂▁

mass

Adults

datasummary(Factor(TEMPERATURE) ~ MASS * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(adult, MASS),  
            fmt = "%.2f")
tinytable_v3db5y85iwpias4vlwt1
TEMPERATURE NUnique mean median min max sd Histogram
27 21 46.93 50.63 29.85 59.93 10.30 ▇▁▃▄▄▆▄
28.5 20 38.81 42.36 16.28 53.26 10.26 ▁▁▄▃▄▇▆▁
30 22 39.94 39.62 23.91 57.31 9.43 ▅▇▂▇▇▇▂▇▅▂

1-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m1_df, MASS.x),  
            fmt = "%.2f")
tinytable_7diugliwp57okxuo9d4n
TEMPERATURE NUnique mean median min max sd Histogram
27 437 0.06 0.06 0.01 0.22 0.03 ▃▇▇▄▂▁
28.5 248 0.07 0.06 0.02 0.19 0.02 ▁▅▇▄▃
30 389 0.07 0.07 0.01 0.15 0.02 ▄▅▇▆▅▃▂▁

2-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m2_df, MASS.x),  
            fmt = "%.2f")
tinytable_235dvgjkqu6huydde6nf
TEMPERATURE NUnique mean median min max sd Histogram
27 460 0.27 0.27 0.03 0.68 0.10 ▁▂▄▇▆▃▁
28.5 297 0.28 0.27 0.06 0.64 0.10 ▁▃▇▇▇▄▁▁▁
30 425 0.28 0.27 0.04 0.62 0.11 ▂▂▅▇▇▄▂▁▁

2.5-months

datasummary(Factor(TEMPERATURE) ~ MASS.x * (NUnique + mean + median + min + max + sd + Histogram), 
            data = drop_na(m2.5_df, MASS.x),  
            fmt = "%.2f")
tinytable_vyg1le2igqtrpr7ckh2k
TEMPERATURE NUnique mean median min max sd Histogram
27 529 0.46 0.46 0.04 1.00 0.17 ▁▂▄▇▇▆▄▁▁
28.5 293 0.45 0.44 0.08 1.00 0.18 ▂▄▆▇▆▅▃▂▁
30 477 0.45 0.45 0.08 0.96 0.15 ▁▂▄▆▇▄▂▁

Fit models [random factors]

1-month

modelNULL <- glmmTMB(MASS.x ~ 1, 
                  family=gaussian(),
                  data =m1_df)

model1 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER), 
                  family=gaussian(),
                  data =m1_df) 

model2 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|LEVEL), 
                  family=gaussian(),
                  data = m1_df)

model3 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER), 
                  family=gaussian(),
                  data = m1_df)

model4 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION), 
                  family=gaussian(),
                  data = m1_df) 

model5 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m1_df) 

model6 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m1_df) 

AIC(modelNULL, model1, model2, model3, model4, model5, model6) 
BIC(modelNULL, model1, model2, model3, model4, model5, model6)

2-month

modelNULL <- glmmTMB(MASS.x ~ 1, 
                  family=gaussian(),
                  data =m2_df)

model1 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER), 
                  family=gaussian(),
                  data =m2_df) 

model2 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|LEVEL), 
                  family=gaussian(),
                  data = m2_df)

model3 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER), 
                  family=gaussian(),
                  data = m2_df)

model4 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION), 
                  family=gaussian(),
                  data = m2_df) 

model5 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m2_df) 

model6 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m2_df) 

AIC(modelNULL, model1, model2, model3, model4, model5, model6) 
BIC(modelNULL, model1, model2, model3, model4, model5, model6)

2.5-month

modelNULL <- glmmTMB(MASS.x ~ 1, 
                  family=gaussian(),
                  data =m2.5_df)

model1 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER), 
                  family=gaussian(),
                  data =m2.5_df) 

model2 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|LEVEL), 
                  family=gaussian(),
                  data = m2.5_df)

model3 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER), 
                  family=gaussian(),
                  data = m2.5_df)

model4 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION), 
                  family=gaussian(),
                  data = m2.5_df) 

model5 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m2.5_df) 

model6 <- glmmTMB(MASS.x ~ (1|CLUTCH_NUMBER) + (1|CLUTCH_ORDER) + (1|REGION) + (1|POPULATION), 
                  family=gaussian(),
                  data = m2.5_df) 

AIC(modelNULL, model1, model2, model3, model4, model5, model6) 
BIC(modelNULL, model1, model2, model3, model4, model5, model6)

For mass measurements at different time periods, including 1, 2, and 2.5 months the best model is the most simply model (i.e., model1), where the only random factor that is present is CLUTCH_NUMBER.

Fit model [fixed factors]

Now that we have figured out which random factors will be included within out generalized linear mixed effects model we can start to explore different hypothesese by adding in our fixed factors - covariates.

Fixed factors that will be included will be those that are essential to answering the initial research question based on heiritability of traits between offspring and parental fish - labelled as MALE and FEMALE in the dataframe as well as their combined score MIDPOINT, if applicable. TEMPERATURE is also essential to answering the main research question that looks to see if heritability changes at different temperatures.

Our main research hypothesis will be modelled using the formula below”

MASS.x ~ MASS_(FE)MALE*TEMPERATURE + (1|CLUTCH_NUMBER)

An alternative research hypothesis will will test will include an interaction with PARENTAL_DAYS_IN_TEMPERATURE to see if heritability was affect by how long adults spent at experimental temperatures. This model may look something like:

MASS.x ~ MASS_(FE)MALE*TEMPERATURE:PARENTAL_DAYS_IN_TREATMENT + (1|CLUTCH_NUMBER)

Lets start fitting models:

offspring-male

1-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m1_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m1_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model1a was selected as the best model and will be used going forward.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.748 0.912 0.944 0.476 0.628 0.504 0.848 0.7 0.264 0.884 0.248 0.532 0.524 0.704 0.464 0.312 0.7 0.544 0.452 0.84 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.058365, p-value = 0.0001575
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0128, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 18, observations = 1387, p-value = 0.04717
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.007709021 0.020432992
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01297765
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.058365, p-value = 0.0001575
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0128, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 18, observations = 1387, p-value = 0.04717
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.007709021 0.020432992
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01297765
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m1_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -7014.2  -6972.3   3515.1  -7030.2     1379 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance  Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.0003148 0.01774 
##  Residual                  0.0003268 0.01808 
## Number of obs: 1387, groups:  CLUTCH_NUMBER, 50
## 
## Dispersion estimate for gaussian family (sigma^2): 0.000327 
## 
## Conditional model:
##                             Estimate Std. Error z value Pr(>|z|)  
## (Intercept)                0.0302112  0.0191034   1.581   0.1138  
## MASS_MALE                  0.0006713  0.0004056   1.655   0.0979 .
## TEMPERATURE28.5            0.0336082  0.0328357   1.024   0.3061  
## TEMPERATURE30              0.0306570  0.0249225   1.230   0.2187  
## MASS_MALE:TEMPERATURE28.5 -0.0005629  0.0007900  -0.713   0.4761  
## MASS_MALE:TEMPERATURE30   -0.0004390  0.0005737  -0.765   0.4441  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                           2.5 %       97.5 %      Estimate
## (Intercept)                       -0.0072308181 0.0676532115  0.0302111967
## MASS_MALE                         -0.0001235985 0.0014661839  0.0006712927
## TEMPERATURE28.5                   -0.0307486790 0.0979650734  0.0336081972
## TEMPERATURE30                     -0.0181902804 0.0795041954  0.0306569575
## MASS_MALE:TEMPERATURE28.5         -0.0021112670 0.0009854722 -0.0005628974
## MASS_MALE:TEMPERATURE30           -0.0015633965 0.0006852967 -0.0004390499
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.0144796866 0.0217422563  0.0177431975
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.516
##      Marginal R2: 0.049

Summary figure

m1.mass <- emmeans(model1a, ~ MASS_MALE*TEMPERATURE, 
                 at =list(MASS_MALE=seq(from =min(m1_df$MASS_MALE), to =max(m1_df$MASS_MALE), by=.25)))

m1.mass.df <- as.data.frame(m1.mass)

m1.mass.obs <- drop_na(m1_df, MASS_MALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m1.mass.obs.summarize <-  m1.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_MALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m1.mass.df, aes(x=MASS_MALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm") + 
  geom_pointrange(data = m1.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MALE MASS (g)") + 
  ylab("OFFSPRING MASS (g)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()
## `geom_smooth()` using formula = 'y ~ x'

2-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(), 
                    data=m2_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m2_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

The null model appears better than the models that we used. Let’s explore the data bit more and see if we can find a reason for this. Let’s start by looking at a basic histogram of our data.

hist(m2_df$MASS.x)

There appears to be a left skew within our data. Let’s see if this can be better modelled with a Gamma distribution. If not we can try to incorporate transformations to our response variable. The model validations below also could use some improving.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.02 0.132 0.056 0.096 0.392 0.284 0.212 0.552 0.18 0.336 0.372 0.724 0.648 0.532 0.132 0.404 0.6 0.252 0.268 0.484 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.03042, p-value = 0.193
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99807, p-value = 0.992
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1263, p-value = 0.7497
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00435550 0.01553001
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.008709422
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.03042, p-value = 0.193
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99807, p-value = 0.992
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1263, p-value = 0.7497
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00435550 0.01553001
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.008709422
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -2285.4  -2244.2   1150.7  -2301.4     1255 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.001475 0.03841 
##  Residual                  0.008888 0.09428 
## Number of obs: 1263, groups:  CLUTCH_NUMBER, 47
## 
## Dispersion estimate for gaussian family (sigma^2): 0.00889 
## 
## Conditional model:
##                             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                0.3139182  0.0490782   6.396 1.59e-10 ***
## MASS_MALE                 -0.0009611  0.0010338  -0.930    0.353    
## TEMPERATURE28.5           -0.0360441  0.0793221  -0.454    0.650    
## TEMPERATURE30             -0.0130914  0.0638280  -0.205    0.837    
## MASS_MALE:TEMPERATURE28.5  0.0010760  0.0018901   0.569    0.569    
## MASS_MALE:TEMPERATURE30    0.0003770  0.0014481   0.260    0.795    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %      Estimate
## (Intercept)                        0.217726762 0.410109676  0.3139182191
## MASS_MALE                         -0.002987347 0.001065120 -0.0009611137
## TEMPERATURE28.5                   -0.191512649 0.119424359 -0.0360441451
## TEMPERATURE30                     -0.138192097 0.112009226 -0.0130914353
## MASS_MALE:TEMPERATURE28.5         -0.002628522 0.004780547  0.0010760123
## MASS_MALE:TEMPERATURE30           -0.002461237 0.003215314  0.0003770389
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.029899653 0.049340932  0.0384093314
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.149
##      Marginal R2: 0.007

Summary figure

m2.mass <- emmeans(model1a, ~ MASS_MALE*TEMPERATURE, 
                 at =list(MASS_MALE=seq(from =min(m2_df$MASS_MALE), to =max(m2_df$MASS_MALE), by=.25)))

m2.mass.df <- as.data.frame(m2.mass)

m2.mass.obs <- drop_na(m2_df, MASS_MALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m2.mass.obs.summarize <-  m2.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_MALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m2.mass.df, aes(x=MASS_MALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m2.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()

2.5-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
BIC(modelNULL, model1a, model1b)

Once again the NULL model seems to outperform our hypothesis testing models. Let’s follow the steps that we conducted for 2-month data and appy a log transformation to our dataset to see if it improved the model.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.936 0.876 0.696 0.428 0.628 0.792 0.84 0.248 0.904 0.272 0.976 0.12 0.684 0.812 0.916 0.66 0.868 1 0.836 0.168 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.030135, p-value = 0.1731
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.96289, p-value = 0.52
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 8, observations = 1347, p-value = 0.5374
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002567476 0.011668738
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.005939124
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.030135, p-value = 0.1731
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.96289, p-value = 0.52
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 8, observations = 1347, p-value = 0.5374
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002567476 0.011668738
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.005939124
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2.5_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -1173.9  -1132.3    595.0  -1189.9     1339 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.005084 0.0713  
##  Residual                  0.022483 0.1499  
## Number of obs: 1347, groups:  CLUTCH_NUMBER, 52
## 
## Dispersion estimate for gaussian family (sigma^2): 0.0225 
## 
## Conditional model:
##                             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                0.3925825  0.0816490   4.808 1.52e-06 ***
## MASS_MALE                  0.0016472  0.0017364   0.949    0.343    
## TEMPERATURE28.5            0.0486000  0.1398557   0.348    0.728    
## TEMPERATURE30              0.0745723  0.1048126   0.711    0.477    
## MASS_MALE:TEMPERATURE28.5 -0.0007853  0.0033636  -0.233    0.815    
## MASS_MALE:TEMPERATURE30   -0.0022103  0.0024207  -0.913    0.361    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %      Estimate
## (Intercept)                        0.232553327 0.552611697  0.3925825116
## MASS_MALE                         -0.001756050 0.005050412  0.0016471811
## TEMPERATURE28.5                   -0.225512062 0.322712097  0.0486000174
## TEMPERATURE30                     -0.130856529 0.280001173  0.0745723223
## MASS_MALE:TEMPERATURE28.5         -0.007377780 0.005807182 -0.0007852992
## MASS_MALE:TEMPERATURE30           -0.006954825 0.002534240 -0.0022102926
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.055841927 0.091045568  0.0713032954
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.192
##      Marginal R2: 0.010

Summary figure

m1.mass <- emmeans(model1a, ~ MASS_MALE*TEMPERATURE, 
                 at =list(MASS_MALE=seq(from =min(m2.5_df$MASS_MALE), to =max(m2.5_df$MASS_MALE), by=.25)))

m1.mass.df <- as.data.frame(m1.mass)

m1.mass.obs <- drop_na(m2.5_df, MASS_MALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m1.mass.obs.summarize <-  m1.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_MALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m1.mass.df, aes(x=MASS_MALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m1.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()

Offspring-female

1-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m1_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m1_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model1a was selected as the best model and will be used going forward.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.812 0.904 0.96 0.528 0.58 0.504 0.892 0.604 0.288 0.872 0.26 0.524 0.556 0.692 0.472 0.276 0.7 0.58 0.46 0.888 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.047398, p-value = 0.005078
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0082, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 17, observations = 1330, p-value = 0.06126
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00746311 0.02038649
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01278195
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.047398, p-value = 0.005078
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0082, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 17, observations = 1330, p-value = 0.06126
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00746311 0.02038649
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01278195
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_FEMALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_FEMALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m1_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -6720.7  -6679.2   3368.4  -6736.7     1322 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance  Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.0002590 0.01609 
##  Residual                  0.0003303 0.01817 
## Number of obs: 1330, groups:  CLUTCH_NUMBER, 48
## 
## Dispersion estimate for gaussian family (sigma^2): 0.00033 
## 
## Conditional model:
##                               Estimate Std. Error z value Pr(>|z|)  
## (Intercept)                  0.0385223  0.0155090   2.484    0.013 *
## MASS_FEMALE                  0.0004705  0.0003347   1.406    0.160  
## TEMPERATURE28.5              0.0130221  0.0241044   0.540    0.589  
## TEMPERATURE30                0.0159863  0.0257814   0.620    0.535  
## MASS_FEMALE:TEMPERATURE28.5 -0.0000554  0.0005596  -0.099    0.921  
## MASS_FEMALE:TEMPERATURE30   -0.0001052  0.0005902  -0.178    0.859  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                           2.5 %      97.5 %      Estimate
## (Intercept)                        0.0081251480 0.068919445  0.0385222965
## MASS_FEMALE                       -0.0001856118 0.001126553  0.0004704707
## TEMPERATURE28.5                   -0.0342216257 0.060265844  0.0130221090
## TEMPERATURE30                     -0.0345443058 0.066516884  0.0159862892
## MASS_FEMALE:TEMPERATURE28.5       -0.0011521572 0.001041365 -0.0000553959
## MASS_FEMALE:TEMPERATURE30         -0.0012620507 0.001051609 -0.0001052208
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.0130535755 0.019838534  0.0160923523
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.474
##      Marginal R2: 0.062

Summary figure

m1_df <-  
  m1_df |> 
  drop_na(MASS_FEMALE)

m1.mass <- emmeans(model1a, ~ MASS_FEMALE*TEMPERATURE, 
                 at =list(MASS_FEMALE=seq(from =min(m1_df$MASS_FEMALE), to =max(m1_df$MASS_FEMALE), by=.25)))

m1.mass.df <- as.data.frame(m1.mass)

m1.mass.obs <- drop_na(m1_df, MASS_FEMALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m1.mass.obs.summarize <-  m1.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.female =mean(MASS_FEMALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m1.mass.df, aes(x=MASS_FEMALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm") + 
  geom_pointrange(data = m1.mass.obs.summarize, aes(x =mean.mass.female, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL FEMALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()
## `geom_smooth()` using formula = 'y ~ x'

2-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(), 
                    data=m2_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m2_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.012 0.096 0.032 0.052 0.344 0.232 0.28 0.436 0.136 0.308 0.376 0.644 0.68 0.484 0.132 0.392 0.608 0.24 0.26 0.48 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.031149, p-value = 0.1902
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99815, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 12, observations = 1212, p-value = 0.4166
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.005126155 0.017231108
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.00990099
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.031149, p-value = 0.1902
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99815, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 12, observations = 1212, p-value = 0.4166
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.005126155 0.017231108
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.00990099
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.012 0.096 0.032 0.052 0.344 0.232 0.28 0.436 0.136 0.308 0.376 0.644 0.68 0.484 0.132 0.392 0.608 0.24 0.26 0.48 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.031149, p-value = 0.1902
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99815, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 12, observations = 1212, p-value = 0.4166
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.005126155 0.017231108
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.00990099
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.031149, p-value = 0.1902
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.99815, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 12, observations = 1212, p-value = 0.4166
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.005126155 0.017231108
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.00990099
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_FEMALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_FEMALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -2206.1  -2165.3   1111.0  -2222.1     1204 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.001354 0.03680 
##  Residual                  0.008810 0.09386 
## Number of obs: 1212, groups:  CLUTCH_NUMBER, 45
## 
## Dispersion estimate for gaussian family (sigma^2): 0.00881 
## 
## Conditional model:
##                               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                  0.2951918  0.0416393   7.089 1.35e-12 ***
## MASS_FEMALE                 -0.0006774  0.0008855  -0.765    0.444    
## TEMPERATURE28.5             -0.0388864  0.0618526  -0.629    0.530    
## TEMPERATURE30                0.0756899  0.0703173   1.076    0.282    
## MASS_FEMALE:TEMPERATURE28.5  0.0013333  0.0014213   0.938    0.348    
## MASS_FEMALE:TEMPERATURE30   -0.0015112  0.0015856  -0.953    0.341    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %      Estimate
## (Intercept)                        0.213580327 0.376803246  0.2951917864
## MASS_FEMALE                       -0.002412839 0.001058091 -0.0006773738
## TEMPERATURE28.5                   -0.160115359 0.082342498 -0.0388864307
## TEMPERATURE30                     -0.062129510 0.213509315  0.0756899028
## MASS_FEMALE:TEMPERATURE28.5       -0.001452316 0.004118892  0.0013332883
## MASS_FEMALE:TEMPERATURE30         -0.004618824 0.001596426 -0.0015111992
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.028369174 0.047737698  0.0368005311
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.149
##      Marginal R2: 0.019

Summary figure

m2_df <-  
  m2_df |> 
  drop_na(MASS_FEMALE)
m2.mass <- emmeans(model1a, ~ MASS_FEMALE*TEMPERATURE, 
                 at =list(MASS_FEMALE=seq(from =min(m2_df$MASS_FEMALE), to =max(m2_df$MASS_FEMALE), by=.25)))

m2.mass.df <- as.data.frame(m2.mass)

m2.mass.obs <- drop_na(m2_df, MASS_FEMALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m2.mass.obs.summarize <-  m2.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_FEMALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m2.mass.df, aes(x=MASS_FEMALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m2.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL FEMALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()

2.5-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_FEMALE*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.948 0.896 0.744 0.5 0.628 0.824 0.828 0.288 0.928 0.34 0.976 0.184 0.66 0.812 0.896 0.716 0.916 1 0.848 0.18 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.033287, p-value = 0.1139
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.95633, p-value = 0.48
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 8, observations = 1293, p-value = 0.6366
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002674849 0.012154598
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.006187162
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.033287, p-value = 0.1139
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.95633, p-value = 0.48
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 8, observations = 1293, p-value = 0.6366
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002674849 0.012154598
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.006187162
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_FEMALE|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_FEMALE * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2.5_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -1147.2  -1105.8    581.6  -1163.2     1285 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.005399 0.07348 
##  Residual                  0.022063 0.14854 
## Number of obs: 1293, groups:  CLUTCH_NUMBER, 50
## 
## Dispersion estimate for gaussian family (sigma^2): 0.0221 
## 
## Conditional model:
##                              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                  0.424324   0.074321   5.709 1.13e-08 ***
## MASS_FEMALE                  0.001028   0.001606   0.640    0.522    
## TEMPERATURE28.5              0.052489   0.115047   0.456    0.648    
## TEMPERATURE30                0.068247   0.118685   0.575    0.565    
## MASS_FEMALE:TEMPERATURE28.5 -0.001076   0.002676  -0.402    0.688    
## MASS_FEMALE:TEMPERATURE30   -0.002163   0.002734  -0.791    0.429    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %     Estimate
## (Intercept)                        0.278657875 0.569990737  0.424324306
## MASS_FEMALE                       -0.002119686 0.004176442  0.001028378
## TEMPERATURE28.5                   -0.172999022 0.277976300  0.052488639
## TEMPERATURE30                     -0.164371146 0.300865760  0.068247307
## MASS_FEMALE:TEMPERATURE28.5       -0.006321504 0.004169887 -0.001075809
## MASS_FEMALE:TEMPERATURE30         -0.007522669 0.003195872 -0.002163398
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.057452294 0.093971308  0.073476985
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.204
##      Marginal R2: 0.009

Summary figure

m2.5_df <- m2.5_df |> 
  drop_na(MASS_FEMALE)
m2.5.mass <- emmeans(model1a, ~ MASS_FEMALE*TEMPERATURE, 
                 at =list(MASS_FEMALE=seq(from =min(m2.5_df$MASS_FEMALE), to =max(m2.5_df$MASS_FEMALE), by=.25)))

m2.5.mass.df <- as.data.frame(m2.5.mass)

m2.5.mass.obs <- drop_na(m2.5_df, MASS_FEMALE, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m2.5.mass.obs.summarize <-  m2.5.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_FEMALE, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m2.5.mass.df, aes(x=MASS_FEMALE, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m2.5.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()

Offspring-Midpoint

1-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(), 
                    data=m1_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family=gaussian(), 
                    data=m1_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model1a was selected as the best model and will be used going forward.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.816 0.908 0.96 0.536 0.584 0.504 0.892 0.604 0.288 0.872 0.268 0.524 0.56 0.696 0.476 0.28 0.708 0.588 0.472 0.892 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.045774, p-value = 0.007594
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0081, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 17, observations = 1330, p-value = 0.06126
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00746311 0.02038649
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01278195
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.045774, p-value = 0.007594
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 1.0081, p-value = 0.872
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 17, observations = 1330, p-value = 0.06126
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.00746311 0.02038649
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                             0.01278195
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MIDPOINT|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MIDPOINT * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m1_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -6721.0  -6679.4   3368.5  -6737.0     1322 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance  Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.0002575 0.01605 
##  Residual                  0.0003303 0.01817 
## Number of obs: 1330, groups:  CLUTCH_NUMBER, 48
## 
## Dispersion estimate for gaussian family (sigma^2): 0.00033 
## 
## Conditional model:
##                                 Estimate Std. Error z value Pr(>|z|)  
## (Intercept)                    0.0329822  0.0166291   1.983   0.0473 *
## MASS_MIDPOINT                  0.0005856  0.0003555   1.647   0.0995 .
## TEMPERATURE28.5                0.0213380  0.0274504   0.777   0.4370  
## TEMPERATURE30                  0.0235153  0.0247728   0.949   0.3425  
## MASS_MIDPOINT:TEMPERATURE28.5 -0.0002358  0.0006499  -0.363   0.7167  
## MASS_MIDPOINT:TEMPERATURE30   -0.0002539  0.0005741  -0.442   0.6583  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                           2.5 %       97.5 %      Estimate
## (Intercept)                        0.0003898950 0.0655745936  0.0329822443
## MASS_MIDPOINT                     -0.0001111277 0.0012822649  0.0005855686
## TEMPERATURE28.5                   -0.0324638480 0.0751397589  0.0213379555
## TEMPERATURE30                     -0.0250384419 0.0720690390  0.0235152985
## MASS_MIDPOINT:TEMPERATURE28.5     -0.0015095584 0.0010378900 -0.0002358342
## MASS_MIDPOINT:TEMPERATURE30       -0.0013790389 0.0008713143 -0.0002538623
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.0130168709 0.0197849204  0.0160479829
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.473
##      Marginal R2: 0.063

Summary figure

m1_df <-  
  m1_df |> 
  drop_na(MASS_MIDPOINT)

m1.mass <- emmeans(model1a, ~ MASS_MIDPOINT*TEMPERATURE, 
                 at =list(MASS_MIDPOINT=seq(from =min(m1_df$MASS_MIDPOINT), to =max(m1_df$MASS_MIDPOINT), by=.25)))

m1.mass.df <- as.data.frame(m1.mass)

m1.mass.obs <- drop_na(m1_df, MASS_MIDPOINT, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m1.mass.obs.summarize <-  m1.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.female =mean(MASS_MIDPOINT, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m1.mass.df, aes(x=MASS_MIDPOINT, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm") + 
  geom_pointrange(data = m1.mass.obs.summarize, aes(x =mean.mass.female, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MIDPOINT STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()
## `geom_smooth()` using formula = 'y ~ x'

2-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(), 
                    data=m2_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family='gaussian', 
                    data=m2_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.012 0.116 0.032 0.052 0.348 0.236 0.292 0.436 0.14 0.324 0.388 0.656 0.688 0.496 0.144 0.404 0.616 0.252 0.268 0.496 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032673, p-value = 0.1503
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.998, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1212, p-value = 0.6256
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.004539118 0.016181158
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.009075908
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032673, p-value = 0.1503
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.998, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1212, p-value = 0.6256
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.004539118 0.016181158
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.009075908
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.012 0.116 0.032 0.052 0.348 0.236 0.292 0.436 0.14 0.324 0.388 0.656 0.688 0.496 0.144 0.404 0.616 0.252 0.268 0.496 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032673, p-value = 0.1503
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.998, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1212, p-value = 0.6256
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.004539118 0.016181158
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.009075908
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032673, p-value = 0.1503
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.998, p-value = 0.984
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 11, observations = 1212, p-value = 0.6256
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.004539118 0.016181158
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.009075908
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MIDPOINT|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MIDPOINT * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -2204.5  -2163.7   1110.3  -2220.5     1204 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.001413 0.03759 
##  Residual                  0.008810 0.09386 
## Number of obs: 1212, groups:  CLUTCH_NUMBER, 45
## 
## Dispersion estimate for gaussian family (sigma^2): 0.00881 
## 
## Conditional model:
##                                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                    0.3011127  0.0459019   6.560 5.38e-11 ***
## MASS_MIDPOINT                 -0.0007978  0.0009697  -0.823    0.411    
## TEMPERATURE28.5               -0.0394936  0.0717508  -0.550    0.582    
## TEMPERATURE30                  0.0339443  0.0690491   0.492    0.623    
## MASS_MIDPOINT:TEMPERATURE28.5  0.0013269  0.0016817   0.789    0.430    
## MASS_MIDPOINT:TEMPERATURE30   -0.0006032  0.0015769  -0.383    0.702    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %      Estimate
## (Intercept)                        0.211146607 0.391078838  0.3011127226
## MASS_MIDPOINT                     -0.002698241 0.001102740 -0.0007977506
## TEMPERATURE28.5                   -0.180122472 0.101135301 -0.0394935859
## TEMPERATURE30                     -0.101389441 0.169277952  0.0339442559
## MASS_MIDPOINT:TEMPERATURE28.5     -0.001969309 0.004623021  0.0013268560
## MASS_MIDPOINT:TEMPERATURE30       -0.003693879 0.002487493 -0.0006031930
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.029040752 0.048651868  0.0375883869
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.150
##      Marginal R2: 0.013

Summary figure

m2_df <-  
  m2_df |> 
  drop_na(MASS_MIDPOINT)
m2.mass <- emmeans(model1a, ~ MASS_MIDPOINT*TEMPERATURE, 
                 at =list(MASS_MIDPOINT=seq(from =min(m2_df$MASS_MIDPOINT), to =max(m2_df$MASS_MIDPOINT), by=.25)))

m2.mass.df <- as.data.frame(m2.mass)

m2.mass.obs <- drop_na(m2_df, MASS_MIDPOINT, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m2.mass.obs.summarize <-  m2.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_MIDPOINT, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m2.mass.df, aes(x=MASS_MIDPOINT, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m2.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MIDPOINT STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()

2.5-month

Models

Main hypothesis
model1a <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)
Alternative hypothesis
model1b <- glmmTMB(MASS.x ~ MASS_MIDPOINT*TEMPERATURE*as.numeric(PARENTAL_DAYS_IN_TREATMENT) + (1|CLUTCH_NUMBER), 
                    family=gaussian(link ="identity"), 
                    data=m2.5_df)

Model selection

AIC(modelNULL, model1a, model1b, k=12) 
## Warning in AIC.default(modelNULL, model1a, model1b, k = 12): models are not all
## fitted to the same number of observations
BIC(modelNULL, model1a, model1b)
## Warning in BIC.default(modelNULL, model1a, model1b): models are not all fitted
## to the same number of observations

Once again the NULL model seems to outperform our hypothesis testing models. Let’s follow the steps that we conducted for 2-month data and appy a log transformation to our dataset to see if it improved the model.

Model validation

DHARMa
model1a |> 
  simulateResiduals(plot=TRUE)  

## Object of Class DHARMa with simulated residuals based on 250 simulations with refit = FALSE . See ?DHARMa::simulateResiduals for help. 
##  
## Scaled residual values: 0.944 0.892 0.728 0.484 0.616 0.812 0.824 0.284 0.928 0.336 0.976 0.176 0.656 0.808 0.896 0.716 0.912 0.996 0.848 0.18 ...
model1a |> testResiduals(plot=T) 

## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032622, p-value = 0.1276
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.95693, p-value = 0.48
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 7, observations = 1293, p-value = 0.4304
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002179302 0.011122433
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.005413766
## $uniformity
## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  simulationOutput$scaledResiduals
## D = 0.032622, p-value = 0.1276
## alternative hypothesis: two-sided
## 
## 
## $dispersion
## 
##  DHARMa nonparametric dispersion test via sd of residuals fitted vs.
##  simulated
## 
## data:  simulationOutput
## dispersion = 0.95693, p-value = 0.48
## alternative hypothesis: two.sided
## 
## 
## $outliers
## 
##  DHARMa outlier test based on exact binomial test with approximate
##  expectations
## 
## data:  simulationOutput
## outliers at both margin(s) = 7, observations = 1293, p-value = 0.4304
## alternative hypothesis: true probability of success is not equal to 0.007968127
## 95 percent confidence interval:
##  0.002179302 0.011122433
## sample estimates:
## frequency of outliers (expected: 0.00796812749003984 ) 
##                                            0.005413766
performance
model1a |> 
  check_model(detrend=FALSE)
## `check_outliers()` does not yet support models of class `glmmTMB`.

Partial effect plots

model1a |> ggemmeans(~MASS_MIDPOINT|TEMPERATURE) |> 
  plot(add.data =FALSE) 

model1a |> ggemmeans(~TEMPERATURE) |> 
  plot(add.data =FALSE) 
## NOTE: Results may be misleading due to involvement in interactions

Model investigation

Summary
model1a |> summary()
##  Family: gaussian  ( identity )
## Formula:          MASS.x ~ MASS_MIDPOINT * TEMPERATURE + (1 | CLUTCH_NUMBER)
## Data: m2.5_df
## 
##      AIC      BIC   logLik deviance df.resid 
##  -1147.3  -1106.0    581.7  -1163.3     1285 
## 
## Random effects:
## 
## Conditional model:
##  Groups        Name        Variance Std.Dev.
##  CLUTCH_NUMBER (Intercept) 0.005385 0.07338 
##  Residual                  0.022063 0.14854 
## Number of obs: 1293, groups:  CLUTCH_NUMBER, 50
## 
## Dispersion estimate for gaussian family (sigma^2): 0.0221 
## 
## Conditional model:
##                                Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                    0.409316   0.079890   5.123    3e-07 ***
## MASS_MIDPOINT                  0.001344   0.001711   0.786    0.432    
## TEMPERATURE28.5                0.053685   0.131473   0.408    0.683    
## TEMPERATURE30                  0.072453   0.115646   0.627    0.531    
## MASS_MIDPOINT:TEMPERATURE28.5 -0.001039   0.003116  -0.334    0.739    
## MASS_MIDPOINT:TEMPERATURE30   -0.002254   0.002692  -0.837    0.402    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova
model1a |> Anova()
Confint
model1a |> confint()
##                                          2.5 %      97.5 %     Estimate
## (Intercept)                        0.252733488 0.565897613  0.409315550
## MASS_MIDPOINT                     -0.002008261 0.004696797  0.001344268
## TEMPERATURE28.5                   -0.203996428 0.311366662  0.053685117
## TEMPERATURE30                     -0.154208910 0.299114069  0.072452579
## MASS_MIDPOINT:TEMPERATURE28.5     -0.007146013 0.005067192 -0.001039411
## MASS_MIDPOINT:TEMPERATURE30       -0.007531169 0.003022221 -0.002254474
## Std.Dev.(Intercept)|CLUTCH_NUMBER  0.057382814 0.093841327  0.073381738
r-squared
model1a |> r2_nakagawa()
## # R2 for Mixed Models
## 
##   Conditional R2: 0.204
##      Marginal R2: 0.010

Summary figure

m2.5_df <- m2.5_df |> 
  drop_na(MASS_MIDPOINT)
m2.5.mass <- emmeans(model1a, ~ MASS_MIDPOINT*TEMPERATURE, 
                 at =list(MASS_MIDPOINT=seq(from =min(m2.5_df$MASS_MIDPOINT), to =max(m2.5_df$MASS_MIDPOINT), by=.25)))

m2.5.mass.df <- as.data.frame(m2.5.mass)

m2.5.mass.obs <- drop_na(m2.5_df, MASS_MIDPOINT, MASS.x) |> 
  mutate(Pred =predict(model1a, re.form=NA, type ='response'), 
         Resid =residuals(model1a, type ='response'), 
         Fit =Pred+Resid) 

m2.5.mass.obs.summarize <-  m2.5.mass.obs |> 
  group_by(CLUTCH_NUMBER, TEMPERATURE) |> 
  summarise(mean.mass =mean(Fit, na.rm=TRUE), 
            mean.mass.male =mean(MASS_MIDPOINT, na.rm = TRUE), 
            sd.mass =sd(Fit, na.rm =TRUE), 
            n.mass = n()) |> 
  mutate(se.mass = sd.mass / sqrt(n.mass), 
         lower.ci.mass =mean.mass - qt(1-(0.05/2), n.mass -1) * se.mass, 
         upper.ci.mass =mean.mass + qt(1-(0.05/2), n.mass -1) * se.mass) |> 
  ungroup()
## `summarise()` has grouped output by 'CLUTCH_NUMBER'. You can override using the
## `.groups` argument.
ggplot(data = m2.5.mass.df, aes(x=MASS_MIDPOINT, y=emmean)) + 
  stat_smooth(aes(color=TEMPERATURE), 
              method = "lm", 
              formula = y ~ x) + 
  geom_pointrange(data = m2.5.mass.obs.summarize, aes(x =mean.mass.male, 
                                                  y =mean.mass, 
                                                  ymin =lower.ci.mass, 
                                                  ymax =upper.ci.mass, 
                                                  color = TEMPERATURE)) +  
  scale_color_manual(values = c("#69d7d8","#ff9c56", "#903146")) + 
  scale_fill_manual(values =c("#69d7d8","#ff9c56", "#903146")) +
  facet_wrap(~TEMPERATURE)+
  xlab("PARENTAL MALE STANDARD LENGTH (mm)") + 
  ylab("OFFSPRING STANDARD LENGTH (mm)") + 
  ggtitle("Offspring-male relationship") +
  theme_classic()